Journal of Mathematical Psychology Mathematical psychology

Journal of Mathematical Psychology 52 (2008) 269–280

Contents lists available at ScienceDirect

Journal of Mathematical Psychology

journal homepage: www.elsevier.com/locate/jmp

Mathematical psychology: Prospects for the 21st century: A guest editorialI

James T. Townsend 1

Indiana University, Department of Psychological & Brain Sciences, 1101 E. 10th Street, Bloomington, IN 47405-7007, United States

a r t i c l e i n f o

Article history:Received 30 April 2007Received in revised form29 April 2008Available online 18 July 2008

Keywords:FutureMathematical psychologyFields of mathematical psychologyHistory of mathematical psychologyPsychological scienceClinical science and mathematicalpsychologyNeuroscience and mathematicalpsychologyMathematical psychology and otherquantitative fieldsComputer science and mathematicalpsychologyPhysics and mathematical psychology

a b s t r a c t

The twenty-first century is certainly in progress by now, but hardly well underway. Therefore, I will takethat modest elasticity in concept as a frame for this essay. This frame will serve as background for someof my hopes and gripes about contemporary psychology and mathematical psychology’s place therein.It will also act as platform for earnest, if wistful thoughts about what might have (and perhaps can still)aid us in forwarding our agenda and what I see as some of the promising avenues for the future. I looselystructure the essay into a section about mathematical psychology in the context of psychology at largeand then a section devoted to prospects within mathematical psychology proper. The essay can perhapsbe considered as in a similar spirit, although differing in content, to previous editorial-like reviewsof general or specific aspects of mathematical psychology such as [Estes, W. K. (1975). Some targetsfor mathematical psychology. Journal of Mathematical Psychology, 12, 263–282; Falmagne, J. C. (2005).Mathematical psychology: A perspective. Journal of Mathematical Psychology, 49, 436–439; Luce, R. D.(1997). Several unresolved conceptual problems of mathematical psychology. Journal of MathematicalPsychology, 41, 79–87] that have appeared in this journal.

© 2008 Elsevier Inc. All rights reserved.

1. Psychology and mathematical psychology therein

1.1. A glimpse of recent history

Psychology is a young science as opposed to a young field

I A word is in order concerning the scope and style of this essay. There aremany themes that could be selected for such a piece. The topic I have chosencenters on the health and future of mathematical psychology. It does include anecessarily very succinct history and a number of bordering issues along with somesuggestions for remediation and improvement. Due to space limitations, I have hadto limit mention and citations of a plethora of important names outside those of avery few of the pioneers in the field. A vast number of important investigators es-pecially if they’re on the younger side are neglected. In particular, I had to virtuallyentirely omit the contributions, many overlapping with mathematical psychology,stemming from the field of psychometrics. The same goes for a voluminous set ofresearches on categorization. I regret and apologize for these necessary omissions.And, though an individual may have performed valuable research in several areasof mathematical psychology, I refrained from making these multiple mentions.I should also mention that I cannot guarantee that all citations within a set ofreferences in the same location in the essay will be homogeneous (e.g., all reviewsof their work, all at about the same date, etc.). It was already laborious simplycollecting these as they are. I also take license to occasionally mention importantnames sans specific citation. The style is meant to be informal, even conversational,so some notable individuals receive mention without specific citation.

E-mail address: [email protected] President of Society for Mathematical Psychology 2004–2005.

0022-2496/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jmp.2008.05.001

of discourse. Ponderings about psychological topics, casual andsystematic, recede into recorded history. Yet we all know thatscientific psychology is little more than one-hundred twenty-five years old or so, dating by convention from 1879, the yearof establishment of Wundt’s famous laboratory in Leipzig; thisaccording to Boring’s (1957) calendar of experimental psychology.

Without question, we have made rather startling progress byalmost any measure, in that scant time. This progress has arguablyaccelerated since the 1940s due to the amazing new tools, soattractive and appropriate for non-physical sciences, proffered byvon Neumann, Wiener, Shannon, and others.2 I speak, of course, ofautomata theory (von Neumann), utility theory (von Neumann &Morgenstern), cybernetics/feedback control theory (Wiener), andinformation theory (Shannon) (see, e.g., Townsend and Kadlec(1990)).

A significant paradigm shift in psychology occurred when itbegan to move away from the grand and perhaps overweeningschools toward less capacious, but more articulated venues con-taining more testable models and hypotheses. The grand schoolswere highly conspicuous in abnormal psychology, especially in the

2 In all cases of name listing, there is no significance to their order.

http://www.elsevier.com/locate/jmp

http://www.elsevier.com/locate/jmp

mailto:[email protected]

http://dx.doi.org/10.1016/j.jmp.2008.05.001

270 J.T. Townsend / Journal of Mathematical Psychology 52 (2008) 269–280

various off-shoots of Freudian theory: for instance, Adler, Fromm,Jung, Horney, and Reich, to name a few. But experimental psychol-ogy also possessed broadly encompassing theories identified withindividuals, as witness Thorndike, Lewin (also featured in socialpsychology), Skinner, Tolman, Hull, and Guthrie.

The 1950s and then the 1960s saw such quantitative areasas signal detection theory, mathematical learning theory, andfoundational measurement theory aid and abet this transition toa less global but more rigorous, science.3 Names we think of in thesignal detection connection are Peterson, Birdsall, Fox (engineers),Swets, Tanner, Green and Egan. Pioneers in mathematical learningtheory include Bush, Mosteller, Estes and again rather quickly,Suppes, Atkinson, Bower, and Crothers, and Murdock.

In addition, just about the same time, psychologists and friendsbegan to work out new theories of measurement that includedstructures more appropriate for the behavioral and biologicalsciences than had the ‘classical’ approaches of Campbell andothers. We note well-known contributors like Suppes, Krantz,Luce, Tversky, and in somewhat different vein, Coombs, andN. Anderson. These investigators also overlapped the growing fieldof decision theory oriented toward actual human behavior. Othernotable names around the same time, in allied pursuits with thesevarious topics were Restle, Greeno, LaBerge, Sternberg, and Theios.Garner and Attneave and Laming helped bring information theoryinto the fold of quantitative psychology. And, the rather neglected(admittedly not quite so much in psychometrics), but fundamentaltopic of geometry in psychology was given a considerable boost byShepard, Kruskal and others.

With the foregoing eye-blink history as preface I move to a listof woes that afflict psychology, afflictions that can be somewhatameliorated by the practice of mathematical psychology.

1.2. The tortoise and the hare

It can prove a frustrating experience to compare psychology’space of advance with progress in the ‘hard’ sciences. Exceptperhaps in the flurry of studies one may witness on discoveryof a dramatic ‘effect’ (see discussion immediately below), stepsin filling in data about a phenomenon not to mention testingof major theoretical issues and models, seem to occur with allthe urgency of a glacier. One may wait years, before a modelerpicks up the scent of an intriguing theoretical problem and carriesit ahead. It is disheartening to contrast our situation with, say,that of microbiology. Perhaps the influx of quantitatively prepared‘outsiders’ (look for later discussion) will helps speed things up.

1.3. The eternal pursuit of effects

One of the ‘blessings and curses’ of modern psychology isthe everlasting quest for ‘effects.’ Some of our most famousinvestigators made their name by discovery of a novel andenchanting effect. In some cases, these have led to a rich set

3 Somewhat truthfully and perhaps somewhat waggishly, the period of the 1960sat Stanford has been referred to, at least by the erstwhile graduate students ofthe times, as “the golden days of mathematical psychology”. Frequent workshopswith other math–psych bastions such as University of Michigan and University ofPennsylvania were held, the milieu was outstanding with a modeling faculty thatincluded William Estes, Richard Atkinson, Gordon Bower and Patrick Suppes. Manyof the students during that period were to become leaders in various facets ofpsychology. The pack in my cohort includes Michael Humphreys, Richard Shiffrin,Don Hintzman, Robert Bjork, Stephen Link, Chiziko Izawa, William Batchelder,Donald Horst, Jack Yellott, Joseph Young, David Rumelhart, and Michael Levine. Inaddition, many already or soon-to-become, notables visited in a post-doctoral orvisiting scientist capacity. With no apologies for the nostalgia, it was a fabulous timeand place to be entering scientific psychology.

of phenomena and interesting, if rarely conclusive, explanatorymodels or theories. A downside is that the careful and steady,incremental growth of the science can be neglected, since theobvious rewards, or at least the ‘grand prizes’ in the field areaccorded the discoveries of effects. But the interpretation of theinitial effect that stands unaltered in the face of replication andparameter variation is quite atypical. Often, follow-up experimentsrequire elaboration or complexification of the original rationaleand frequently, yet more experimentation. Thus, this approachcan be productive if carefully pursued experimentally andtheoretically. Otherwise, our priorities can be decidedly skewedtoward unearthing of new effects.

1.4. Another portentous trade-off approach: Operationalism

A close cousin of the ‘effects’ bias is the embodiment of one ormore theoretical notions in an experimental paradigm, or set ofparadigms. The growth of ‘operationalism’ was associated with thephilosophical movement of logical positivism and the Vienna Circle(think Schick, Ayer, Carnap, Neurath, Hempel and influencers andfellow travelers such as Russell, Wittengenstein, Popper) and founda vent in psychology through efforts of Meehl & MacQuordale andthe physicist Bergmann (it has been said that Bergmann influencedthe social sciences far more than he did physics). Again, thisapproach possesses benefits, and aided psychology in shaking offthe residue of impossible-to-answer philosophical conundrumsthat still adhered to the field in the early twentieth century. Theloved and hated school of behaviorism came into existence anddominated experimental psychology for several decades.

In any event, there were snags, snags which were proba-bly unanticipated by the founders of the ‘operational definition’wherein philosophical claptrap was avoided by defining theoret-ical entities by way of the empirical operations through whichobservations were recorded. A substantial snag is the risk of cir-cularity where a theoretical hypothesis points to an experimentalresult and vice versa. The theoretical and phenomenal restrictionsare evident. Or, if as sometimes occurs, there are several more-or-less distinct operations relating to a phenomenon and presumedtheoretical concept, these may turn out to be closely related (iden-tical occasionally, this is good), unrelated, or even contradictory,depending on a subsequent theory developed to encompass theparadigms associated with respective operational definitions.

Mathematical psychology serves as a stiff antidote to the af-flictions of ‘effect-philia’ and cul-de-sacs of overindulged oper-ationalism. The necessity of providing a rigorous, economical,accounting of concepts and empirical through a quantitative modelclearly combats the overly particular, and acts not only to ac-commodate an entire set of phenomena, but assays the abil-ity of diverse theoretical notions and experimental operations to‘live together’ within the same theory.

1.5. Anti-replication and anti-null effects bias

With regard to afflictions in psychology (and probably a numberof other sciences), I mention two which are not particularlysolvable through modeling, but I want to get them off my chest:One is the heavy bias against replication. Too few studies arepublished that precisely replicate an earlier study. In fact, youngerinvestigators in particular, are warned to include variations so thateditors and reviewers will not reject their ‘replication’ out of hand.This is not an earmark of a mature science. Long ago, William K.Estes spoke of this regrettable prejudice in a seminar during mygraduate student days. It seems just as true now as it did then.

A cognate weakness that many writers have remarked on,using a variety of terms, is the bias against publication (or evenconsideration for publication) of ‘null’ results. How on earth can

J.T. Townsend / Journal of Mathematical Psychology 52 (2008) 269–280 271

psychology advance when only ‘positive’ findings with regard to ahypothesis, model, etc., are made available to the scientific public?A number of suggestions have been put forth to circumvent thisobstacle, including an archive of negative findings, perhaps sortedaccording to topic. To my knowledge, this strategy has never beenseriously attempted. With the current revolution in electronicpublishing, it might be worth a try. With the increasing abilityof information systems to implement content-addressable search,even just the compilation of frequencies of positive vs. negativefindings could be implemented in the drawing of inferences.

Aside from null results, recent years have seen a lively debateabout the value of null hypothesis testing, in addition to variousmeans of improving the strategy and avoiding its threateningsloughs of despond.

1.6. Methodology = Modeling plus statistics and psychometrics

Another associated topic involves methodology in psychology.A multitude of writers have decried the methodological fettersaccompanying decades of over-reliance on standard statistics,statistics originally intended more for assessing grain productionunder varying conditions than for an evolving systematic science.On the one hand, traditional statistics and its offspring have servedas effective tools for much of the advancement of psychologicalscience in the last century. On the other, an argument can bemade that we have become much too dependent on its fruits inways that have succored the tendency to ‘live-with’ loose, verbaltheories. Done right, the field of mathematical psychology offers aprescription for using quantitative theory to impel theory-drivenmethodology.

Of course, there will always be a need for statistics. In additionto testing various traditional hypotheses, providing for confidenceintervals, etc., it is highly useful to have in hand tools to test modelfits and, even better, to compare models against one another (see,e.g., the useful special issues of Journal Mathematical Psychology(Myung, Forster, & Browne, 2000; Wagenmakers & Waldorp,2006)). This topic will reappear in later sections of the presentessay.

1.7. A regrettable neglect of proper scientific referencing

I will conclude this section with a little belly aching pertinentto the behavioral sciences in general and psychology in particular.Obviously, we all know of areas in psychology that are amenableto modeling and could clearly profit from it. Often, verbally based“theories” may be underpinned (or even contradicted) throughpreviously published quantitative research but the verbal theoristfails to use or cite it. It may well be that the ‘errant’ investigatorthought up the approach independently in a verbal and qualitativeway, as psychologists often do. Or it may be that they simply don’tfeel that anything accomplished mathematically has to be cited bynon-modelers, as noted above. In either case, it is bad science andinsulting to the theorists and methodologists who labor mightilyto advance psychology as a rigorous science. I trust that a physicistwould not fail to recognize mathematical results that formed asturdy basis under her theory or tools.4

A related gripe concerns the tendency of investigators indifferent quantitative, but overlapping, disciplines to ignore workin the areas of their research ‘cousins’. Sometimes this simplytakes the form of failing to utilize helpful material from otherresearch domains, but not infrequently, it may involve lapses in

4 I suspect a number of readers can think of instances of this phenomenon,perhaps intersecting their own work. I rather cravenly fail to mention names inorder to avoid litigation and other unpleasant retribution.

citing even mathematical theorems that have been previouslypublished elsewhere. Mathematical psychology is occasionallyprey to this tendency since we overlap such a broad sweepof mathematically oriented research venues. The writer haswitnessed several examples of these oversights. Economics,industrial engineering, operations research, biophysics, manyareas of artificial intelligence (robotics, pattern recognition,problem solving . . . ) clearly provide for rich interaction amonginvestigators but are likewise prone to this type of abuse.

2. Mathematical psychology: Rumination on its evolution andposition

2.1. Continued development of areas of mathematical psychology andrumors of a demise

As observed earlier, mathematical psychology emerged fromembryo in the late fifties and early sixties of the twentieth century.Already by the end of the sixties, some were pronouncing thedemise of mathematical psychology.5 Did it really die, despite theostensible continued existence of practitioners of that field? If notan obvious corpse, is it in dire peril?

Let’s pause a moment to espy the fate of some of the majorareas of early effort in mathematical psychology mentioned earlier.Perhaps they may shed some light on this assertion. In addition,this little tour will allow us to discern where major branches of thefield have themselves perambulated.

2.2. Foundational measurement

Several of the founders’ names were mentioned earlier but itwould be remiss not to cite several major seminal works in thisarea: The three foundational volumes of Krantz, Luce, Suppes, andTversky (1971), Luce, Krantz, Suppes, and Tversky (1990a,b) andSuppes, Krantz, Luce, and Tversky (1989), the excellent pedagogicaltext of Roberts (1979), and the seminal topologically based volumeof Pfanzagl (1968).

Foundational measurement theory has continued to attracthighly skilled mathematicians and quantitatively oriented psy-chologists from throughout the world and to advance knowledge,especially on the critical topic of ‘meaningfulness’. Technical dis-cussion is beyond my scope here, but informally, ‘meaningfulness’relates jointly to how a measurement scale represents qualitativeaspects of the real world, and the degrees and types of invariancethat properties of a scale enjoy under permitted transformations ofthat scale (e.g., Krantz et al. (1971) and Roberts (1979); for a recentthoughtful statement, see Narens (2003)).

One impediment to more usage of foundational measurementtheory has undoubtedly been the relative paucity of effort and re-sults on an ‘error’ theory which could provide a ready implemen-tation of statistical procedures with data. Groundbreaking workcontinues on this challenge (for recent progress on stochastic ap-proaches to foundational measurement which subsume the tra-ditional error theory, see, e.g., Niederée and Heyer (1997) andRegenwetter and Marley (2001)).

I believe this field is and will continue to be, of interest not onlyto the behavioral and biological sciences, but also to philosophyof science and epistemology in physics, although at present thesubstrata primarily relate to Newtonian rather than relativisticphysics. At any rate, this branch of mathematical psychology isapparently not responsible for the reputed passing of the field.

5 As far as I can ascertain, these announcements have been confined to verbalremarks. However, they have occasionally been uttered by renowned psychologists.


2.3. Signal detection theory

Signal detection theory, as most readers of this essay willknow, emerged in the 1950s as a confluence of Neyman/Pearsonstatistical decision theory and ideal detector theory of electricalengineering (e.g., one of my early favorites is Peterson, Birdsall,and Fox (1954)). In psychology, the Green and Swets (1966) bookhas become a classic with other volumes such as Egan (1975)following up on interesting byways. Early on, there were clearand continuing associations to be made with Thurstone’s populardiscriminal processes theory.

Signal detection theory went through a period during the sixtiesand early seventies in which psychologists proffered a numberof models that lay outside the engineer-oriented mathematicalcommunication theory (as exemplified by the central content ofthe popular (Green & Swets, 1966)), the latter also incorporating(but was not limited to) elementary statistical decision theory.In addition to Luce’s adaptation of his choice theory to detectionand recognition situations (e.g., Luce (1959, 1963a)), a number offinite state models were put forth and evaluated (e.g., Atkinsonand Kinchla (1965), Krantz (1969) and Luce (1963b)). Althoughthese contained interesting capabilities, not always captured bythe dominant theory (as exemplified by Green and Swets (1966)),especially learning effects (e.g., Kinchla, Townsend, Yellott, andAtkinson (1966) and Luce (1959)), they currently see onlyoccasional employment.

In contrast, the dominant theory is now ubiquitous inpsychology in general as well as in sensory sciences, usuallyemploying the ubiquitous Gaussian distributions (nonetheless,two-state models vie with the continuous theory in certainareas; see the Wixted reference below and citations therein).With regard to the latter characteristic, it would seem thatcertain foundational results could be beneficially employed byexperimental researchers (e.g., Marley (1971)). For example,ideal detector theory is flourishing as are multidimensionaltheories of signal detection. An example of the latter is generalrecognition theory, which was originally developed in order tostudy interdimensional interactions (e.g., Ashby and Townsend(1986), Kadlec and Townsend (1992) and Thomas (1999, 2003)).One of the major contributions of signal detection theory wasthe centrality of decision mechanisms, even in putative “purelysensory” domains. This facet continues to offer fresh perspectivesas in the Wenger and Ingvalson (2002) study which found apowerful role of the decision process in the perception of holisticvs. non-holistic object perception. It has been widely employed asa theory of categorization (e.g., Ashby and Maddox (1990); manyof the models in Ashby (1992) are signal detection based).

Certainly, signal detection theory should be studied (usuallyit is covered rather cursorily, if at all, and in a non-quantitativefashion–more is the pity; see below) even by students in theirintroductory courses. It stands as the prototypical theory-drivenmethodology, since it can be employed to discern decision andlearning bias from ‘true’ sensory or sensitivity (e.g., signal-to-noise ratio) effects in such diverse fields as hypnotic phenomena,to trial-witness memory, to laboratory psychophysics or learningand cognition experiments e.g., Swets (1996); see Balakrishnan(1999) for an alternative method of analyzing sensory and biaseffects). For instance, Wixted (2007) argues for a ‘traditional’ typeof signal detection model against less mathematized but processoriented, two-process kinds of models, in certain areas of cognition(e.g., see Balota, Burgess, Cortese, and Adams (2002), Diana, Reder,Arndt, and Park (2006), Heathcote (2003), Hockley and Cristi(1996), Malmberg, Zeelenberg, and Shiffrin (2004) and Yonelinas(1994)). MacMillan and Creelman (2005) compile a worthy setof signal detection-based methodologies provided in a tutorialstyle. Wickens (2002) provides a basic introduction to many of thefundamental concepts.

In any event, there is no reason to think that signal detectiontheory encouraged anyone to proclaim the decease of mathemati-cal psychology.

2.4. Decision theory

What about the field of decision making? One branch ofeffort primarily theoretical but impelled partly by a growingliterature of experimentation, finds its roots in the axiomaticfoundations laid down by von Neumann and Morgenstern (1953).Psychologists dedicated to this tradition consisted partly of thosealso contributing to foundational measurement, and indeed, oftensimilar tools are found in their theoretical arsenal (e.g., Luce,Suppes, Krantz, and Tversky). Of course, this branch also includedthose with statistics or economics backgrounds (e.g., Savage).

A massive and influential development in the field came aboutthrough the efforts of Tversky and Kahneman, who discovereda number of human choice situations in which people veerdrastically away from the classic (and even some newer) axiomatictheories. Certain of these cases flow from their theoretical results,especially prospect theory (Kahneman & Tversky, 1979). Aseveryone should now know, this corpus of work earned Kahnemanthe Nobel Prize in economics and would undoubtedly beenawarded simultaneously to Tversky were it not for his untimelypassing.

The field of decision making has historically been somewhatseparated into a set of quantitative theorists (mostly in theaxiomatic or statistical framework) and a set of experimentalists,the latter largely made up of psychologists. Of course, there stillare many who do both (e.g., One subdivision of the experimentalgroup has inclined toward testing predictions made by theaxiomatic or statistically-based models (Birnbaum, 2004). Anotherarea has concentrated on continuing the quest for psychologicalbehavior that seems at odds with various facets of the axiomatictheories [especially utility theory]). Yet another has evolvednon-quantitative models or theories that attempt to be heavilyreal-world oriented (e.g., how do people make decisions in agroup-crisis?), and experimented thereon.

Although various facets and extensions of utility theory perse are still active research areas, a number of investigators havemoved on to explore the preferential laws governing risk (Weber,Shafir, & Blais, 2004). In addition, the perhaps overdue appearanceof hedonics in the consequences of decisions has occurred (Mellers,2000). Gigerenzer and colleagues have been a powerful voicein plumping for models based on all-too-human limitations inprocessing capacity and sometimes rationality (e.g., Gigerenzerand Todd (1999)). Wishing to avoid charges of excessive modesty,I hasten to mention work that seeks to be in the spirit of strongquantitative theory but heavily invested in psychological andbiologically flavored knowledge (Busemeyer & Townsend, 1993).6In any event, I can locate little in the evolving field of decisionmaking that should have precipitated augurs a terminal malady ofmathematical psychology.

2.5. Psychophysics

I somewhat artificially separated signal detection from psy-chophysics and it might legitimately be argued that neither isstrictly a subfield of mathematical psychology. Moreover, due tospace and ‘psychological distance’ concerns, I must neglect the nowsizeable field of sensory sciences, which of course, heavily over-laps both psychophysics and signal detection (and uses methods

6 I refrain from mentioning the likelihood of a pummeling about my head andshoulders by my colleague, Jerome Busemeyer.


from both). Nonetheless, I think it is fair to say that in some sensesthe psychophysics of Weber, Fechner (to a lesser extent Wundt),Helmholtz, and others was a progenitor, along with statistics, ofmathematical psychology. Also, modern mathematical psycholo-gists and psychophysicists have long been attracted to it, apply-ing concepts from measurement theory and functional analysis(e.g., see Falmagne (1985) and Luce, Bush, and Galanter (1963)) aswell as more process models (e.g., Baird (1997) and Link (1992)).

From some viewpoints, Stevens (1951) should probably be con-sidered as the twentieth century heir of classical psychophysics,through his innovation of the necessity for a hierarchy of mea-surement scale types as well as his new experimental methods ofscaling, the so-called direct methods. One influential investigatorwhose work could be listed in several of the present categories in-cluding the present is that of Shepard (1964). Another is Anderson(1981). A substantial and evolving theory of psychophysics withroots in Fechner’s original developments is found in the work ofDzhafarov and Colonius (e.g., Dzhafarov (2002) and Dzhafarov andColonius (2007)). Nosofsky has contributed an influential modelingapproach which links up psychophysics, psychometrics (via multi-dimensional scaling), and information processing (e.g., Nosofsky(1984)).

An enduring challenge, not unrelated to the issue of multidi-mensional interactions mentioned earlier, is the inclusion of con-text effects in psychophysical scaling (see, e.g., Helson (1964) andParducci (1956)). The drawing together of the psychophysical ap-proaches with process model approaches to context and inter-actions has begun (e.g., Chapter 16 in Baird (1997), Hughes andTownsend (1998) and Link (1992)) but is likely still in its infancy.Nonetheless, it appears that such a synthesis can be useful. Forinstance, a coalition of psychophysical with process methodol-ogy has elucidated aspects and challenges of the Stroop effect notpreviously visible (Melara & Algom, 2003). Sarris has made funda-mental contributions to relational psychophysics in comparative-developmental psychology (e.g., Sarris (2006)).

2.6. Neural modeling

One very significant player in quantitative theory in psychologyas well as cognitive science has been neuropsychological modeling.It might have been expected to provide a major counter weight tosome of the less propitious forces facing mathematical psychology.Due to the potentially symbiotic linkages that could result whenboth are applied in the spirit of reductionism (but see Uttal (1998)),plus its almost startling resurgence in recent years, I take a littlemore space on this topic.

Like some of the other branches of research, an adequatehistory of neural modeling would take up at least one volume.However, we can limn in some of the most evident sign-posts.Although neural modeling was certainly around at the time thatmathematical psychology received its formal impetus, and therehave been significant intersections over the years, it is not routinelythought of as a sub-field of mathematical psychology.

Modern history of neural modeling goes back at least toRashevsky and Ashby in the 1940s. The fifties saw emergence oflogic network based thinking (flowing undoubtedly from automatatheory of von Neumann, Turing, and others) by McCulloch and Pitt.The ultimate influence of Hebb’s well-known principles of synapticlearning, though not so rigorously presented originally as someof the other candidates for attention, would be difficult to overemphasize.

At the juncture where mathematical learning theory began tofade in popularity (no causal implication intended here, merely atime-marker), Grossberg (1969) began to publish his first neuralmodeling work which, incidentally, included an emphasis onlearning and motivation. That work was, and is, founded on

non-linear (typically multiplicative) differential equations, mostoften with strong systems-oriented interpretations afforded theequational elements.

About this time too, two investigators in the cognitive sciencerevolution, Minsky and Papert (1969), published a book innocentlyentitled “Perceptrons”. As most readers of this essay are wellaware, perceptrons are basically linear, and usually deterministic,pattern recognizers; members of the class of systems known aslinear discriminant classifiers. These authors showed in a carefullyreasoned treatise, that perceptrons are incapable of seeminglyquite elementary topological distinctions.

Needless to say, their exegesis did not have the effect offurthering the general interest in perceptron theory or perhaps,even an encouragement with regard to neural modeling per se. Infact, some have felt that it had a rather devastating consequenceon activity in neural modeling.

During the years that interest in neural modeling paled,Grossberg and a few other theorists, such as James Anderson,kept the fires aflame (Anderson, 1995). Anderson’s models couldbe viewed as more sophisticated upgrades (e.g., incorporatingstochastic noise, Hebbian learning devices, and non-linear decisionstructures) of perceptronic tenets.7

Then, in the 1980s, hurtled the connectionistic meteor, drivenin substantial part by the labors of Rumelhart and McClelland(1986). Interestingly, Rumelhart had been a graduate studentin the Stanford mathematical psychology training program inthe 1960s. Even though some specialists would prefer to offerdistinct definitions for “connectionism” vs. say, “distributedprocessing”, they are often used interchangeably to indicate whatwe might call neuralistic modeling. “Neuralistic” might be areasonable neologism for this field since practitioners strive tolet neurophysiology and neuro-anatomy guide their efforts whilebenignly neglecting less critical aspects of these disciplines. Theyare occasionally taken to task for transgressions in this regard,but theorization has always been a matter of emphasizing whatseems most vital and ignoring the rest. Certainly such criticismsapply to all but the most microscopic models of neural functioning;and the latter are often of little interest to psychologists. It is alsoworth remarking, given our earlier discussion, that learning theory,always a key ingredient of neural modeling, made an impressivecome back in the wake of the renaissance of the latter.

Although endeavors were made by those involved with theSociety for Mathematical Psychology to encourage affiliations andinteractions (e.g., by appointing connectionistic associate editorsto Journal of Mathematical Psychology; inviting keynote addressesby leaders in connectionistic modeling, etc.), and although manymathematical psychologists have labored from this perspective,the attempts to embrace this field perhaps have not been entirelysuccessful. There are notable exceptions, including the work ofKruschke (1992).

Neuropsychological modeling has certainly waxed and wanedover the past half century or so, and has proceeded more or lessindependently of mathematical psychology, but it doesn’t appearto have been responsible for the obituary of the latter at any pointin time.

2.7. Information processing approach

Like the other topics discussed here, the information processingapproach possesses somewhat fuzzy boundaries, but perhaps

7 In the discipline of neural modeling even more than others, space and thenatural emphasis of this essay prohibit listing of a sizeable set of investigatorsprimarily associated with other fields, which have made fundamental contributionsto neuro-psychological quantitative theorizing.


even more so. A central tenet seems to be the representationof perceptual, cognitive, and/or motoric mechanisms, usuallyconfined to a certain task setting, via a set of subsystems withthe information flow (itself usually a fuzzy concept) depicted viathe so-called, and ubiquitous, flow diagram. Lachman, Lachman,and Butterfield (1979) provided what I regard as a quintessentialtutorial on the information processing approach, and one of the lastrigorous treatments of cognitive psychology for undergraduates. Ofthe early textbooks in mathematical psychology, probably Laming(1968) was the closest to the information processing approach.Undoubtedly spawned by computer science (e.g., automatatheory), and aided and abetted by information theory andcybernetics, the generic conception was picked up just as fast bygeneral experimental psychologists as by mathematical modelers,sans the rigor of the latter. Yet, the seminal efforts of Sternberg,Sperling, Estes, Falmagne, Atkinson, and others, helped attractmodelers intrigued by the idea of analyzing a system down intoits functional components, even if (or because) the propertiesof the system in action might reveal emergent properties. Latercontributors of the modeling ilk include Nosofsky, Massaro, Link,Yellott, Ratcliff, D. Meyer, Colonius, Diederich, J. Miller, Vorberg,Bundesen, E. A. C. Thomas, Massaro, Logan, Schweickert, Dzhafarovand others.

Models of response time have been especially influential andproductive in uncovering underlying processing mechanisms. Themost popular types of response time models, often includingaccuracy predictions have been those based on random walks (e.g.,Link and Heath (1975)), diffusion processes (e.g., Ratcliff (1978)and Busemeyer and Townsend (1993)), and counting processes(e.g., Smith and Van Zandt (2000)).

I believe the essence of the information processing approachprovided a milieu that helped prompt investigations into issuesof model mimicking. This influence was even felt in modeling oflearning and memory. Thus, Greeno and Steiner (1964) analyzedmodel equivalence within Markov chain models of memory.Batchelder (1970) attacked what seemed to be a mimicking issueregarding incremental vs. all-or-none learning and showed howto distinguish them. My own efforts on parallel vs. serial modeltesting began shortly thereafter (e.g., Townsend (1969, 1971,1972) and Townsend and Wenger (2004)). Of course, knowledgeof when and how models cannot be differentiated can assist inerecting a meta-theory or methodology that is capable of suchassay (e.g., Townsend (1976a,b)). In my not-unbiased view, theinformation processing approach is still alive and prospering,although perhaps not always under that rubric.

2.8. Mathematical learning theory

And then there is that other early pillar of mathematicalpsychology, mathematical learning theory. The germinating workof Bush and Mosteller (1955) and Estes (1950) gave rise to a decadeor so of fervent activity on mathematical models of learning,including a wave of anthologies and monographs (e.g., Bush andMosteller (1955); later, Atkinson, Bower, and Crothers (1965); verylate, Restle and Greeno (1970)).

Even during the 1960s, mathematical learning theory wasbeginning to move toward more emphasis on memory and lesson learning. By the mid-seventies publications of theoretical andexperimental effort devoted to mathematical learning theory haddiminished precipitously. Furthermore, it could be (and was,at least informally) argued that finite-state Markov learningmodels, a mainstay in the field, were beginning to appear ratherbaroque, sometimes without convincing signals from the data thatsuch elaborations were required. Perhaps before self-correctivemeasures could eventuate, other forces essentially moved in tooccupy the territory.

Thus, by this time, the budding field of cognitive science, withboth its theoretical content as well as implementation heavilydetermined by digital computers and automata theory, was in-creasing enormously in popularity, with Simon and Newell per-haps leading the charge in areas close to experimental psychology.Contributions began pouring in not only from experimental psy-chologists but also philosophers, computer scientists, electrical en-gineers and applied physicists. Such innovations as productionsystems soon provided a powerful lure to psychologists seeking aricher milieu for concepts about mental operations. Likely, it is pri-marily the confluence of cognitive science as a new and promisingfield along with the perceived failure of mathematical learning the-ory to ‘pay off’ that caused a substantial hiatus, if not termination,of the latter. Into the bargain, many of the founders of mathemati-cal learning theory were increasingly attracted to regions of studymore closely allied with cognitive science, such as the broad ap-proach of human information processing. It seems fair to say thatthe well-known Atkinson and Shiffrin model of short-term mem-ory and control processes (Atkinson & Shiffrin, 1968) epitomizedthis movement. The subsequent launching of the ACT models byJ. R. Anderson and colleagues (e.g., Anderson and Bower (1973))in some ways intersect both the information processing approachand the emerging cognitive science paradigm (e.g., a la Simon &Newell).

The evidence seems to point to the recession of mathematicallearning theory in the late 1960s as one key marker thatconvinced some investigators of the weak health, if not demise,of mathematical psychology. Moreover, a number of modelerswho cut their teeth with the ‘classical’ learning models, werenow moving wholesale into ever broader and more rigorousmodels of memory. I refer in particular, to the works of Hintzman(1986), Izawa (1971), Murdock (1982), (e.g., Raaijmakers andShiffrin (1981)), (e.g., Humphreys, Bain, and Pike (1989)), whichinvestigators have over the past few decades made this field into aposter child of how behavioral research should be carried out.

Okay, so we could lay some of the blame for the perhaps pre-mature rumor of the death (but not interment!) of mathematicalpsychology on the adventitious events pertaining to learning the-ory in the late 1960s. However, it has always seemed paradoxicalto me how learning per se was shoved into the corner for sev-eral decades, even though memory became supersedent as a le-gitimate topic in cognitive science and mathematical psychologyper se. How did these memories originally become instated? Withfew exceptions only in the relatively lonely (for awhile!) terrain ofneural modeling (e.g., Anderson (1973) and Grossberg (1969); andshortly with a resurgence from the Anderson quarter e.g., Ander-son (1990)), did learning theory persevere. While it is fair to saythat learning played some part in some of the burgeoning mem-ory models, it was at best a minor role. Learning as a legitimateresearch topic is happily now back with us big time.

In any event, it is time to abandon this historical, if somewhatwhimsical, excursion and pose the query as to the state of healthof the field today.

2.9. Trends, education, politics, and tenure

One can attempt answers along many dimensions. First, the“glass is half full” perspective: The Society of MathematicalPsychology continues to serve many functions that are conducivenot only to the highest standards of research in mathematicalpsychology, but to encourage the entering of young scientistsinto our field. Its members contribute expert reviewership to abroad spectrum of research areas vis-à-vis journals and scientificgrant proposals. They provide some of the top research, especiallythrough model and theory building, in scientific psychology. Manyregularly bring in valuable grant resources to their universities,


even in these extreme financially harsh times. Many members ofthe Society have served on strategic committees and panels innational institutes and other societies and federations.

Nonetheless, there are serious “glass is half empty” concerns:For instance, with reference to membership of Society of Math-ematical Psychology, we find that unfortunately, numbers have,with some fluctuation, tended to decline in number since its for-mation in the late 1960s (following inauguration of Journal of Math-ematical Psychology in 1964). We will see that there is widespreadanxiety about quantitative training in psychology, not simply in-clusive of mathematical psychology per se. We can question ‘why’and ask about relationships to various movements in psychologyand cognate areas.

2.10. Fractionation vs. generality

One trend over the past century up to the present has been thesteady fractionation of science in general and certainly psychologyin particular. This trend has been associated with many benefits,such as deepening knowledge and specialization within manybranches of study. Yet, it has arguably, if rather paradoxically, beenat the expense of areas like mathematical psychology which seek toencompass a wide sweep of individual domains, from learning andmemory, to sensation and psychophysics (indeed to at least onesociety for each modality!), to models of social behavior, to neuraltheory. As these specialties have budded and flowered, generalisticgroups such as SMP have often experienced hurdles in capturingtime from over-extended researchers.

2.11. Is psychology now ‘sold’ on mathematical psychology, havingeffectively absorbed it?

One argument heard occasionally, even by ‘friends’ of mathe-matical psychology, is that now quantitative modeling has beenabsorbed by the field at large and hence a separate sub-disciplinedevoted to this specialty is superfluous. For instance, serious ob-stacles in the path of publishing mathematical models encouragedthe establishment of Journal of Mathematical Psychology. Support-ing evidence is proffered that many experimental journals nowregularly accept quantitative modeling of the experimental datawithin submitted articles. I agree that this is definite evidence ofprogress. Yet, the acknowledgement of the fact does not lead inex-orably to the consequent. For one thing, journal editors reveal anastonishingly high variance with regard to their attitudes towardquantitative modeling. And, many if not most, outside of quantita-tive journals, pre-form some type of upper-bound on the degree ofabstraction they deem acceptable.

My view is that import alone should determine what seesthe light of day even in our more experimental or qualitativeorgans, but with the stipulation that the editor should be free torequest fairly considerable clarification and even tutorial material.A rebuttal might be to the effect that “Well that just means morepublications for the Journal of Mathematical Psychology”. A problemwith this reasoning is that if there is little or no overlap withthe experimental journals, it is even easier for less quantitativescientists to ignore our work.

I think it can be fairly argued that SMP and Journal ofMathematical Psychology, in addition to sister organizations andjournals such as Psychometric Society and Psychometrika, aregreatly needed by scientific psychology as “keepers of the flame”.We serve as upholders of the highest quantitative standards byour multitudinous duties as reviewers of papers and grants and aspractitioners of modeling and methodological science. All this, inaddition to our function in training of graduate and undergraduatepsychology majors.

2.12. Undergraduate training: Following the trends in society

Many surveys and studies have documented the steep declineof scientific training and acumen in our youth, not to mention thedeterioration of scholarship in general (e.g., witness the lamentablepruning of elementary and high school courses in music, foreignlanguages, and even physical education, starting in the 1970s).Hence, it should come as no surprise to learn that scientificeducation of US college students is woefully inadequate.

With the possible exception of some well endowed private uni-versities and colleges, psychology departments serve as bountifulcash cows for their institutions, even (or especially) in large, publicresearch-oriented universities. In addition to sizeable flocks of ma-jors plus non-majors, they typically bring in large amounts of out-side research monies, sometimes close to or exceeding the moreestablished laboratory science departments. And, their classroomlab facilities cost little to nothing in comparison with the latter.

The debt to the devil in all of this is that there is immensepressure, if usually implicit, against driving down enrollments byincreasing standards, for instance, by requiring majors to takemore physical and biological science and mathematics. Of course,this influence sums with many others, including the aversioncertain sectors and individuals within psychology feel towardsmathematics and hard science.8 Other pernicious forces includethe seemingly perpetual inclination of publishers to persuadeauthors to ‘dumb down’ their textbooks and sometimes, evenscientific monographs, and the well-documented grade inflationthat has plagued higher education at least since the advent of the1970s.

With regard to undergraduate training, given the aboveand other factors, expecting a sea change toward solid-scienceeducation, even for most psychology majors, not to mention thelegions from other departments who take our courses, is akin tobelief in the tooth fairy. The only practical solution I can espy isfor psychology departments to offer a true scientific psychologytrack, with mandatory courses in the sciences, mathematics andstatistics. It could, but need not, be incorporated into a true honorsprogram.9

The latter could include options for coursework in engineering,economics, ecological sciences, and more recently available,informatics and biocomplexity, which would increase the chancesof those who decide not to pursue postgraduate education, to findemployment. An added benefit to psychology graduate programsthroughout the land, would be a diminution in the usual scenarioevery spring: thirty or so departments fighting over a pitiablysmall handful of qualified students, the latter of which eithercome from other sciences or somehow manage to acquire decentbackground in the face of feeble departmental curricula andinadequate counseling.10 Certainly, the uninterrupted flow of milk

8 Naturally, much of this aversion is muted, especially overt statements ofopposition to “hard science” in general. However, one could not get far, evenin rigorous neuroscience, in the absence of a modicum of real mathematics(e.g., the calculus). In addition, shocking as it may seem, occasionally first-handanecdotes surface from individuals in major universities, of prominent researchpsychologists disparaging quantitative methodology and training (and this includespsychometrics and statistical methodology; not just mathematical modeling).

9 It has appeared to me that many departments and universities began todowngrade the standards associated with honors programs back in the 1960s and70s. The reasons may vary but at least in some cases, one motive was to avoiddamage to self esteem—a noble goal, but perhaps at odds with the definition of“honors”.

10 Who among us has not heard some version of the following undergraduate’srefrain (and for me across several universities where I’ve taught): “But the counselorlaughed when I asked about taking math and science, and asked why a psychologistwould need something like that”. Of course, the one saving grace has been therelatively continuous influx of quantitatively prepared students from abroad.However, even this palliative may fade as countries begin to establish their ownfine universities and research institutes.


from the contented cow should gladden the hearts of deans andpresidents.

2.13. Graduate recruitment and training

Given this prelude, perhaps it should not come as a surprisethat even graduate training in psychology has seen the devolutionof quantitative training; never that auspicious in the first place.Several prominent psychometricians reported and discussed asurvey on quantitative training of psychologists in 1990 (Aiken,West, Sechrest, and Reno (1990); cf. Townsend (1994)). At thattime outdated topics were a major concern with a consequentunder representation of newer, powerful strategies.

Fifteen years later we have a reprise on a considerably morefrightening note, in an article by Clay (2005) written for theAPA Monitor: the shocking scarcity of new quantitatively-trainedpsychologists, and even of qualified programs to train them.11

Stephen West (Editor of Psychological Methods and Professor atArizona State University) is quoted “At lot of the major quantitativeprograms over the years have died. We’re one of three largerprograms in psychology in the country, and we produced one Ph.D.this year”.

Although mathematical psychology per se, probably never sawmore than three or four formal programs of training in the US,it is still arguable that even that limited presence has declined.Of course, the dearth of rigorously trained post-baccalaureatepsychologists mentioned above, and the degradation of science ingeneral in the US, are undoubtedly contributing factors.

Now, the Clay article portrays the view from some methodol-ogists that quantitative psychologists are in great demand, withsome institutions finally ‘giving up’ when years go by without asuccessful quantitative hire. That may be and if so, I applaud thatthere is now a palpable appreciation of our specialty, without, ofcourse, hearty clapping for the diminution in numbers. I must saythough, that in my opinion, years went by where superbly trainedmathematical and psychometric psychologists were not accordedthe best job opportunities. Too often, the unspoken refrain seemedto be something of the form “. . . well, everyone learns statistics ata sufficient level to teach our introductory courses [often the onlykind offered], so we might as well hire in one of our favorite con-tent areas and be doubly happy . . . ”. And, a rather dangerous pitfalloften awaited young quantitative psychologists, that of being ex-pected to provide ‘free’ consulting services to faculty and students(who often didn’t bother to take the assistant professor’s classeson the topic), yet receiving little credit for these activities at tenureand promotion time.

As other possible aversions to the beginning graduate student,in addition to the sheer arduous technical preparation (years ofmathematics plus the applied quantitative tools), it has appeared

11 Partly to lodge support for this Monitor article and partly to stress thecontributions of mathematical psychology to quantitative training in psychology,I co-authored with Richard Golden and Thomas Wallsten, a guest editorial on thisissue for the APA Science Directorate (Townsend, Golden, & Wallsten, 2005). Thetopic of quantitative training in psychology was raised several times in various‘break-out’ sessions at the recent APA sponsored Science Leadership Conference.Admittedly, APA has in the past been accused of short-changing the ‘science’ infavor of professional concerns, undoubtedly one of the several motivations forthe establishment of American Psychological Society. Nonetheless, there is someevidence that APA, in addition to its decided interest in professional matters, iscredibly moving into the game of promoting scientific psychology. For instance, Iwas recently able to recommend ‘challenges in quantitative training in psychology’as a theme for the next leadership conference. Steve Breckler, Executive Director forScience, APA, appears to provide a healthy force in this direction (see PsychologicalScience Agenda (see [email protected], for more on APA’s role in psychologicalscience). In fact, APA is in the process of forming a task force on quantitative trainingin psychology and hopefully such movements can help turn the tide.

to me that it is the very rare neophyte (i.e., newly mintedquantitative assistant professor) who can publish at the rateeasily achieved by their cohort in other specialties of the field.Some departments, T&P committees and significant individuals(especially pertinent is the chair or department head) doappear to ‘handicap’ according to sub-discipline when consideringpromotions and salaries, but I think this is not common and whendone, probably not to the appropriate degree.

Why, it might be countered, does the burgeoning field ofcognitive neuroscience seem to have little difficulty in attractingapprentices? It is true that a huge range of technical abilityand background is accommodated within that discipline but thesame could be true in quantitative psychology. More persuasively,neurophysiology has been a more entrenched part of psychology— indeed, almost every department has an ‘area’ devoted toneuroscience or neurocognition (the “in” terms for this region,‘physiological psychology”, “psychobiology”, and so on, seem tochange every decade or so) — since the very inception of scientificpsychology. There are also feeder sources and ancillary trainingposts for the physiologically inclined, like pre-med and biologyfor which there is little concomitant in our area. Interestingly,the current movement toward neuroscience forms one of the fewtrends in psychology toward hard science.

In any event, it can be cogently argued that the centraladvantage psychology has over other fields in years past has beenthe relatively heavy component of education in practical statisticsand methodology and potentially, modeling. Hundreds of hoursof arduous, often boring and occasionally exciting, labor in thelaboratory plus the subsequent data analysis and model testing,put psychologists in a solid position not only for academic positionsbut also research and management in industry and government.

The pivotal role of experimental and methodological psychol-ogists in effectively leading medical research teams especially inexperimental design and data analysis, while typically serving un-der the obligatory M.D principal investigator, is well known. And,clinical psychologists have neither the political clout, M.D. prestige,nor monetary recompense afforded psychiatrists. But, they havein the past been able to contribute their knowledge and practiceof test theory and administration. These skills largely disappearin the unfortunate trend toward so-called Psy.D. degrees, whichrequire little if any research experience, statistical knowledge oreven training in psychological test theory. However, there are nowforty-five member training institutions in clinical science, whichadhere to the principles of the Academy of Psychological ClinicalScience. The goal of these programs is to emphasize the rigoroustraining of a core of clinical scientists (see, e.g., McFall (2006)) per-haps somewhat compensating for an avalanche of Psy.D. practi-tioners over the past couple of decades. I don’t know if researchhas been accomplished regarding the relative proficiency in ther-apy of Psy.D. personnel vs. traditional Ph.D. clinical psychologists,but there is no doubt about their respective statistical and researchskills. Although as one would expect, one of the major regions ofcross disciplinary training for clinicians is in neuro-science, somedepartments, such as that at Indiana University, train some stu-dents in mathematical modeling and collaborate with the quanti-tatively oriented faculty.

Many of us have witnessed even people with Ph.D.s in suchfields as physics, engineering, computer science, and mathematics,making grave errors in experimental design when they ‘cross-over’ into experimental psychology in the absence of collaborationwithin the latter.12 The same is true, and more, with regard to

12 This is not a criticism of these groups. Who would expect even a superblytrained mathematical psychologist to do the work of a physicist, chemist, orengineer?


competition of our clinicians with psychiatrists in the ability toorganize, run and analyze research programs, and of course, togive and interpret psychologically based assessments. In addition,at present, unless they also have had significant training inexperimental psychology, especially in laboratory work, they oftenseem to be deficient in the ‘lore’ of what solid psychologicalresearch is all about.

Will mathematical psychology perish? In fact, will psychologyas constituting the leading body of investigators into psychologicalresearch issues, shrivel and fade? Unfortunately, I think neither ofthese grim results can be ruled out. As already noted, outstandingscientists from the disciplines mentioned earlier, are beginning toattack research problems in regions formerly thought to be ourprovince, in numbers almost meriting the term “flood”. No foolsthey, it is actually surprising that we’ve had the field to ourselves solong. They collectively bring an armamentarium of research toolsthat we could well wish our own matriculating psychologists weremastering.

Naturally, it follows that the science itself will not die, not evenquantitatively oriented theory and theorists. Only it would be alittle bit sad if psychology as a formal discipline, not to mention ourown mathematical psychology, both with their venerable history,should fail to continue as the core of mental science.

Let us turn finally turn to what I hope are some furtherinteresting challenges as well as actions in which we can engagethat can lead to a more promising morrow.

3. Denouement of the present: The future

What are some hurdles that must (or at least should) beovercome in order to optimize our science? What can we do toremain vibrant and even grow over, say, the next fifty years? Thesetwo goals may be thoroughly intertwined. Some possibilities thatoccur to me follow below. Some are continuances of pathwaysalready opened and no doubt there are many more that will beoffered and implemented in the time to come.13

1. First, I make mention of three barriers in modeling: A. Thechallenge of distinct theories which predict the same majordata corpora, and even become more structurally alike as theystrive to encompass new and old data. This barrier is oftenfound when a subject matter has been probed by models andexperimentation over a number of years. A classic example,even within the sphere of ‘less mathematically precise’ is foundin the comparison of Tolman’s quite centralistic theory ofbehavior vs. Hull’s more behavioristic (both, of course, becameknown as “neo-behaviorists”, and we hasten to reiterate thatHull made more effort at mathematization of his concepts).This may be happening within certain well-studied areassuch as long-term memory (doubtlessly, there will be somedisagreement here) and perhaps to some extent, within areasof the categorization literature. Of course, new data andexperiments can continue to yield fruit, but there is a dangerof new studies becoming increasingly ‘precious’ as to thedecreasing consequence of the new phenomena. One way tobreak this cycle may be the import of neuro-imaging and otherphysiologically based strategies. B. The challenge of modelcomplexity. Ever since the dawn of cognitive science, therehas been the obstacle posed by extremely complex models.Jokes have often been made to the effect that one might as

13 Naturally, we hereby enter irrevocably the portals of “do as I say not as I do”,since I could well have toiled more assiduously myself on many of these paths.Nonetheless, hopefully even if most of us attempt to ‘pay our due’ in a small sectorof these, the cumulative effect will be mighty.

well be studying the human brain as opposed to trying to figureout what a very complicated model is predicting, or especiallyhow it is predicting it. Although this problem originally rearedup within ‘traditional’ AI type models (e.g., those basedon seemingly interminable computer programs in LISP), itreappeared in the embryonic connectionism. Even relativelysimple models with hidden units could be rather inscrutable.Modern connectionist models tend to take several tacksin negotiating these ‘rapids’: a. Hidden layers are assignedspecific roles to play in a cognitive task. b. Naturally, the tried-and-(maybe) true usual statistical techniques such as factoranalysis, etc. are brought to bear on the black box system. c.Theoretical lesions (a non-black box approach) are employedto discern the various mechanisms. It has also been claimedthat so-called ‘local connectionist’ models are less prone tothis vein of identifiability problem than ‘globally distributed’models (e.g., see interesting discussions in Grainger andJacobs (1998)).14 C. Even rather simple experiments canrequire more data than is reasonable to acquire. For instance,general recognition analysis of various types of independenceworks optimally when a factorial design is employed with a1–1 stimulus–response assignment. Thus, a five dimensionalexperiment with four levels on each dimension would demand45 or 1024 stimuli. Only 100 trials per condition (we liketo run at least 300/condition), precipitates over one hundredthousand trials. At 500 trials per day (quite an arduousschedule), this would consume approximately ten months.There had better be a darn good chance of accumulatingsolid meaningful data, for a very good theoretical reason,to justify such an experiment. Of course, the time honoredtack here is to run multiple subjects. This would still be inmost cases, unpractical to say the least. More theoreticallyproblematic is the traditional practice of subject averaging,now acknowledged almost universally to be flawed not onlyin process modeling (e.g., Ashby, Maddox, and Lee (1994)and Estes (1956)) but also in psychometrics, where it isomnipresent (see Molenaar (2004)). Nonetheless, obeying theprecept that (almost) nothing is categorical, we have some ‘justin’ findings which indicate that for small data samples, fittingmodels to subject averages across is sometimes superior toindividual fits (Cohen, Sanborn, & Shiffrin, in press).

2. Mathematical psychologists and psychometricians should joinforces to strengthen and broaden quantitative training ofpsychologists. Perhaps for historical reasons, there seems tohave been reluctance by parties of both sides to do muchof this in the past. Inexplicably, commissions on quantitativetraining in psychology have sometimes omitted to involverepresentatives of mathematical psychology. Perhaps thepresent crisis will encourage more cooperation (even if of theform “If we don’t all hang together, we shall certainly all hangseparately”).

3. We should encourage the construction and strengtheningof undergraduate tracks which possess higher standardsand upgraded, more scientifically oriented and quantitativelydependent, class material. Even though these would probablybe expressed as something of the form, “. . . for honors-levelpsychology majors . . . ” they would be open to capable studentsfrom mathematics, as well as the physical and biologicalsciences.

4. Psychology should require more undergraduate and gradu-ate courses in mathematics and the physical sciences. Perhapspsychology departments should offer more elementary math-ematics courses integrated into psychological statistics and

14 I’m indebted to John Kruschke for an enlightening discussion of these matters.


modeling in the same spirit as do the various fields of engi-neering. Outside of the usual very low level statistics coursesrequired of everyone, these might be reserved for the under-graduate students meeting the standards mentioned in (2)above. Although we will never stand independent of coursesfrom mathematics, building more of our own courses and re-quiring at least some of them possesses a number of advan-tages. These include: A. Uniform training in various topics.B. Affords an opportunity to employ examples from the so-cial and biological sciences, especially scientific psychology.This tactic renders the material more interesting and convinc-ing and helps lead into the applications in advanced psychol-ogy courses. C. Instills an esprit de corps (admittedly perhapssomewhat like the “we’re all in boot camp together” phe-nomenon) which helps bond budding psychologists. D. Actingas ‘feeder courses’ into more advanced quantitative methodol-ogy and modeling training.

Of course, we face not only opposition from certain sectorsof our own discipline but the usual territoriality of otherdepartments which teach similar material (again, engineeringhas surmounted this common challenge and, presumably, socould we). Another challenge, maybe especially pervasive ingraduate echelons, is the often oppressive number of generalcore courses required of graduate students. However, thistype of challenge is somehow circumvented by psychologicalneuroscientists so I don’t see it as insurmountable.

5. My opinion is that we have to be willing to teach statistical andgeneral methodological courses within our own department.15

It is a lamentable fact that there is significant pressure toteach these at what we consider to be a criminally low level.However, in most college and university settings, the numberof ‘slots’ accorded any sub-discipline will be at least partly,and sometimes completely, a function of service teachingresponsibilities. We will almost certainly wither as a field ifwe cannot justify our contributions partly through teachingcommitments.

6. Our thinking on the exploding interest by applied physicists,computer scientists, applied mathematicians must be divertedfrom “here come the (perhaps ‘at least as smart as us’)barbarians”, to a more positive outlook.16 We should try tobring them into collaborations and perhaps mutual teachingassignments, perhaps in advanced courses or seminars. Wecould encourage the most interested to attend SMP, and otherscientific psychology conferences. Our students attend theircourses, so perhaps we could lure some of their studentsto take ours. The burgeoning field of cognitive science andrelated disciplines, both in academia and industry, certainlyoffer attractive job prospects for some excellently trainedscientists (e.g., high-energy physics and mathematics) whosejob prospects may not be too rosy these days, unfortunately.And, it would jointly aid them and our science, not to mentionquantitatively sound research, if they were to receive moretraining in our content research areas.

7. Related to but not identical to (5), practitioners of thenew fields of brain imaging are hungry for tools andtraining pertaining to methodology and data analysis of theirdata. We could make a real contribution to their statistics

15 Not everyone in quantitative psychology agrees with this declaration,as witnessed by several conversations I’ve had with colleagues recently atconferences; thus the ‘opinion’ clause.

16 Intriguingly, not all of this interest in deriving from what in the past havebeen called “applied” mathematics. A few years ago, I attended a workshopat a respected mathematical society on the topic of learning machines whichincluded presentations by several topologists, not often thought of, even by othermathematicians, as especially ‘applied’.

(e.g., time series; parameter estimation; hypothesis testing)and engineering (e.g., systems identification) approaches toname a couple. In addition, this is one of a number ofareas where statistics and substantive process modelingcould synthesize with the neuroscience to offer vigorousinstruments for progress. Especially rich opportunities lie inthe relative scarcity of means of comparing and provisionallylinking two or more types of data, for instance, behavior,fMRI, EEG, PET, single unit recordings, and so on. Ashby’s(e.g., Ashby, Alfonso-Reese, Turken, and Waldron (1998))recent modeling ventures into such regions appear prophetic.These movements are going to be of enormous consequence inthe years to come.17

8. There is a small, but significant and hopefully growingpresence of rigorous process modeling in clinical psychologicalscience. One of the earliest pioneers has been Richard W.J.Neufeld (e.g., see his new edited volume, (Neufeld, 2007);papers include Neufeld (1993) and Neufeld, Vollick, Carter,Boksman, and Jette (2002). Other innovative examples includeStout, Rock, Campbell, Busemeyer, and Finn (2005) and Treat,McFall, Viken, and Kruschke (2001). In point of fact, there is asector of clinical science where the researchers are as ‘tough-minded’ as any in cognitive science, as and perhaps more opento mathematical modeling than some of the latter. It wouldbehoove us to welcome and offer our ‘aid and abetting’ to thissmall but growing field.

9. Simple finite-state process models, most often with a Markovassumption transiting between states, have been with usfor quite a spell (e.g., finite-state signal detection modelsmentioned earlier). Batchelder has assembled a very generaltheory-driven methodology and related statistical armamen-tarium based on this concept, which he terms cognitive psycho-metrics. These constituent models differ from traditional linearand log-linear models by virtue of their tree structure.

10. In addition to mathematical psychology’s time honored abilityto test parameterized models against experimental data, Iwon’t miss a chance to plug approaches that test entireclasses of models against one another in ways that areinvariant over specific distributions and parameterizations.Such theory-driven methodologies (I like the term ‘meta-theory’ or ‘meta-modeling’ for such approaches), have beenespecially prominent and successful in identification ofmental architecture and accordant mechanisms in responsetimes (e.g., Schweickert, Giorgini, and Dzhafarov (2000) andTownsend and Wenger (2004)) and featural and dimensionalindependence in accuracy (e.g., Ashby and Townsend (1986)and Kadlec and Townsend (1992)).

11. In the 1970s and beyond, mathematical psychology finallybegan to build models capable of handling both accuracy aswell as response times (e.g., Link and Heath (1975) and Ratcliff(1978)). Yet, for decades our models and most of our data havebeen relatively confined to n = 2 stimuli and responses. It isimportant both for the basic science as well as applicationsin many fields to extend our theories and methodologies tosizeable values of n for stimuli and responses.

12. In the last decade or so, a number of mathematical psychol-ogists have played major roles in plowing virgin territory inmodel testing (e.g., see the special recent JMP issue on modelselection guest edited by Wagenmakers and Waldorp (2006)and the slightly older JMP issue edited by Myung et al. (2000).

17 Speaking personally, I have been ‘having a ball’ in the last couple of years, nowthat we have a T-3 fMRI machine at IU, interacting not only with neuroscientists,but also with the physicists and others either helping run and maintain or simplybeing attracted to this new technology.


Some of this work, pioneered in psychology by Myung and col-leagues, exploits novel concepts from complexity theory andpermits for the first time, model testing to take into accountthe ability of a model to account for a wide range of data, apartfrom the sheer number of parameters. These new vistas cannot only aid our own field, but help us to aid colleagues fromother fields, such as mentioned in (6) just above.

In closing, I suppose it’s considered rather outré’ to sound likea cheerleader or NFL coach, but I do hope that each of us cando something, in addition to our personal research, to make acontribution to our field. Mathematical psychology has arguablyaccelerated the evolution of psychology and allied disciplines intorigorous sciences many times over their likely progress in itsabsence. Let’s nurture and strengthen it.

References

Aiken, L. S., West, S. G, Sechrest, L., & Reno, R. R. (1990). Graduate training instatistics, methodology, and measurement in psychology: A survey of Ph.D.programs in North America. American Psychologist, 45, 721–734.

Anderson, J. A. (1973). A theory for the recognition of items from short memorizedlists. Psychological Review, 80, 417–438.

Anderson, N. H. (1981). Foundations of information integration theory. New York, NY:Academic press.

Anderson, J. R. (1990). The adaptive character of thought. Hillsdale, NJ: LawrenceErlbaum Associates.

Anderson, J. A. (1995). An introduction to neural networks. Cambridge, MA: MIT Press.Anderson, J. R., & Bower, G. H. (1973). Human associative memory. Hillsdale, NJ:

Lawrence Erlbaum Associates.Ashby, F. G. (1992). Scientific psychology series, Multidimensional models of perception

and cognition. Hillsdale, NJ: Lawrence Erlbaum Associates.Ashby, F. G., Alfonso-Reese, L. A., Turken, A. N., & Waldron, E. M. (1998).

A neuropsychological theory of multiple systems in category learning.Psychological Review, 105, 442–481.

Ashby, F. G., & Maddox, W. T. (1990). Integrating information from separablepsychological dimensions. Journal of Experimental Psychology: Human Perception& Performance, 16, 598–612.

Ashby, F. G, Maddox, W. T., & Lee, W. W. (1994). On the dangers of averaging acrosssubjects when using multidimensional scaling for the similarity choice model.Psychological Science, 5, 144–151.

Ashby, F. G., & Townsend, J. T. (1986). Varieties of perceptual independence.Psychological Review, 93, 154–179.

Atkinson, R. C., Bower, G. H., & Crothers, E. J. (1965). Introduction to mathematicallearning theory. New York, NY: Wiley.

Atkinson, R. C., & Kinchla, R. A. (1965). A learning model for forced-choice detectionexperiments. British Journal of Mathematical and Statistical Psychology, 18,87–123.

Atkinson, R. C., & Shiffrin, R. M. (1968). A proposed system and its control processes.In K. W. Spence, & J. T. Spence (Eds.), The psychology of learning and motivation:Advances in research and theory: Vol. 2. New York, NY: Academic Press.

Baird, J. C. (1997). Scientific psychology series, Sensation and judgment: Complemen-tarity theory of psychophysics. Mahwah, NJ: Lawrence Erlbaum Associates.

Balakrishnan, J. D. (1999). Decision processes in discrimination: Fundamental mis-representations of signal detection theory. Journal of Experimental Psychology:Human Perception & Performance, 25, 1189–1206.

Balota, D. A., Burgess, G. C., Cortese, M. J., & Adams, D. R. (2002). The word-frequencymirror effect in young, old, and early-stage Alzheimers’ Disease: Evidencefor two processes in episodic recognition performance. Journal of Memory &Language, 46, 199–226.

Batchelder, W. H. (1970). An all-or-none theory for learning on the paired-associateand concept levels. Journal of Mathematical Psychology, 7, 97–117.

Birnbaum, M. H. (2004). Causes of allais common consequences paradoxes:Unexperimental dissection. Journal of Mathematical Psychology, 48, 87–106.

Boring, E. G. (1957). History of experimental psychology. New York, NY: Appleton-Century-Crofts.

Busemeyer, J., & Townsend, J. T. (1993). Decision field theory: A dynamic-cognitiveapproach to decision making. Psychological Review, 100(3), 432–459.

Bush, R. R., & Mosteller, F. (1955). Stochastic models for learning. New York, NY:Wiley.

Clay, R. (2005). Too few in quantitative psychology. Monitor on Psychology, 26–28.Cohen, A. L., Sanborn, A. N. & Shiffrin, R. M. Model evaluation using grouped or

individual data. Psychological Bulletin & Review (in press).Diana, R. A., Reder, L. M., Arndt, J., & Park, H. (2006). Models of recognition: A review

of arguments in favor of a dual process account. Psychonomic Bulletin & Review,13, 1–21.

Dzhafarov, E. N. (2002). Multidimensional Fechnerian scaling: Pairwise compar-isons, regular minimality, and nonconstant self-similarity. Journal of Mathemat-ical Psychology, 46, 583–608.

Dzhafarov, E. N., & Colonius, H. (2007). Dissimilarity cumulation theory andsubjective metrics. Journal of Mathematical Psychology, 51, 290–304.

Egan, J. P. (1975). Signal detection theory and ROC analysis. New York, NY: AcademicPress.

Estes, W. K. (1950). Toward a statistical theory of learning. Psychological Review, 57,94–107.

Estes, W. K. (1956). The problem of inference from curves based on group data.Psychological Review, 53, 134–140.

Estes, W. K. (1975). Some targets for mathematical psychology. Journal ofMathematical Psychology, 12, 263–282.

Falmagne, J. C. (1985). Elements of psychophysical theory. London, Oxford: UniversityPress.

Falmagne, J. C. (2005). Mathematical psychology: A perspective. Journal ofMathematical Psychology, 49, 436–439.

Gigerenzer, G., & Todd, P. M. (1999). Simple heuristics that make us smart. New York:Oxford University Press.

Grainger, J., & Jacobs, A. M. (Eds.) (1998). Localist connectionist approaches to humancognition. Mahwah, NJ: Erlbaum Associates.

Greeno, J. G., & Steiner, T. E. (1964). Markovian processes with identifiable states:General considerations and application to all-or-none learning. Psychometrika,29, 309–333.

Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. NewYork, NY: Wiley.

Grossberg, S. (1969). Embedding fields: A theory of learning and physiologicalimplications. Journal of Mathematical Psychology, 6, 209–239.

Heathcote, A. (2003). Item recognition memory and the receiver operatingcharacteristic. Journal of Experimental Psychology: Learning, Memory, & Cognition,29, 1210–1230.

Helson, H. (1964). Adaptation level theory. New York, NY: Harper.Hintzman, D. L. (1986). Schema abstraction in multiple-trace memory model.

Psychological Review, 93, 429–445.Hockley, W. E., & Cristi, C. (1996). Tests of the separate retrieval of item and

associative information using a frequency-judgment task. Memory & Cognition,24, 796–811.

Hughes, H. C., & Townsend, J. T. (1998). Evidence for independent ON and OFFchannels in human vision. Psychological Science, 9, 53–60.

Humphreys, M. S., Bain, J. D., & Pike, R. (1989). Different ways to cue a coherentmemory system: A theory for episodic, semantic, and procedural tasks.Psychological Review, 96, 208–233.

Izawa, C. (1971). The test trial potentiating model. Journal of MathematicalPsychology, 8, 200–224.

Kadlec, H., & Townsend, J. T. (1992). Signal detection analysis of multidimensionalinteractions. In F. G. Ashby (Ed.), Probabilistic multidimensional models ofperception and cognition. Hillsdale, NJ: Lawrence Erlbaum Associates.

Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision underrisk. Econometrica, 47, 263–299.

Kinchla, R. A., Townsend, J., Yellott, J. I., Jr., & Atkinson, R. C. (1966). Influenceof correlated visual cues on auditory signal detection, perception, andpsychophysics. Perception & Psychophysics, 1, 67–73.

Krantz, D. H. (1969). Threshold theories of signal detection. Psychological Review, 76,308–324.

Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971). Foundations ofmeasurement: Vol. 1. New York, NY: Academic Press.

Kruschke, J. K. (1992). ALCOVE: An exemplar-based connectionist model of categorylearning. Psychological Review, 99, 22–44.

Lachman, R., Lachman, J. L., & Butterfield, E. C. (1979). Cognitive psychologyand information processing: An introduction. Hillsdale, NJ: Lawrence ErlbaumAssociates.

Laming, D. (1968). Information theory of choice reaction times. New York: AcademicPress.

Link, S. W. (1992). The wave theory of difference and similarity. Hillsdale, NJ: LawrenceErlbaum Associates.

Link, S. W., & Heath, R. A. (1975). A sequential theory of psychologicaldiscrimination. Psychometrika, 40, 77–105.

Luce, R. D. (1959). Individual choice behavior. New York, NY: Wiley.Luce, R. D. (1963a). Detection and recognition. In R. D. Luce, R. R. Bush, & E. Galanter

(Eds.), Handbook of mathematical psychology: Vol. 1. New York, NY: Wiley.Luce, R. D. (1963b). A threshold theory for simple detection experiments.

Psychological Review, 70, 61–79.Luce, R. D. (1997). Several unresolved conceptual problems of mathematical

psychology. Journal of Mathematical Psychology, 41, 79–87.Luce, R. D., Bush, R. R., & Galanter, E. (1963). Handbook of mathematical psychology:

Vol. 1. New York, NY: Wiley.Luce, R. D., Krantz, D., Suppes, P., & Tversky, A. (1990a). Foundations of measurement:

Additive and polynomial representations: Vol. 1. New York, NY: Academic Press.Luce, R. D., Krantz, D. H., Suppes, P., & Tversky, A. (1990b). Foundations of

measurement: Representation, axiomatization, and invariance: Vol. 2. San Diego,CA: Academic Press.

MacMillan, N. A., & Creelman, C. D. (2005). Detection theory: A users guide. Mahwah,NJ: Lawrence Erlbaum Associates.

Malmberg, K. J., Zeelenberg, R., & Shiffrin, R. M. (2004). Turning up the noise orturning down the volume? On the nature of impairment of episodic recognitionmemory by Midazolam. Journal of Experimental Psychology: Learning, Memory, &Cognition, 30, 540–549.

Marley, A. A. J. (1971). Conditions for the representation of absolute judgment andpair comparison isosensitivity curves by cumulative distributions. Journal ofMathematical Psychology, 8, 554–590.

McFall, R. (2006). Doctoral training in clinical psychology. Annual Review of ClinicalPsychology, 2, 21–49.


Melara, R. D., & Algom, D. (2003). Driven by information: A tectonic theory of Stroopeffects. Psychological Review, 110, 422–471.

Mellers, B. A. (2000). Choice and the relative pleasure of consequences. PsychologicalBulletin, 126, 910–924.

Minsky, M., & Papert, S. (1969). Perceptions. Cambridge, MA: MIT Press.Molenaar, P. C. M. (2004). A manifesto on psychology as idiographic science:

Bringing the person back into scientific psychology, this time forever.Measurement, 2(4), 201–218.

Murdock, B. (1982). A theory for the storage and retrieval of item and associativeinformation. Psychological Review, 89, 609–626.

Myung, I. J., Forster, M. R., & Browne, N. W. (Eds.) (2000). Model selection. Journal ofMathematical Psychology, 44(1), 1–232 (special issue).

Narens, L. (2003). Theories of meaningfulness. In S. Link, & J. T. Townsend (Eds.),Scientific psychology series. Mahwah, NJ: Erlbaum Associates.

Neufeld, R. W. J. (1993). Stochastic modeling of stimulus encoding and memorysearch in paranoid schizophrenia: Clinical and theoretical implications. In R.L. Cromwell (Ed.), Schizophrenia: Origins, processes, treatment, and outcome: Thesecond Kansas series in clinical psychology. (pp. 176–196). New York, NY: OxfordUniversity Press.

Neufeld, R. W. J. (Ed.) (2007). Advances in clinical cognitive science. Washington, DC:APA press.

Neufeld, R. W. J., Vollick, D., Carter, J. R., Boksman, K., & Jette, J. (2002). Applicationof stochastic modeling to the assessment of group and individual differences incognitive functioning. Psychological Assessment, 14, 279–298.

Niederée, R., & Heyer, D. (1997). Generalized random utility models in therepresentational theory of measurement: A conceptual link. In A. A. J. Marley(Ed.), Choice, decision and measurement: Essays in honor of R. Duncan Luce(pp. 155–189). Mahwah, NJ: Erlbaum Associates.

Nosofsky, R. M. (1984). Choice, similarity, and the context theory of classification.Journal of Experimental Psychology: Learning, Memory, and Cognition, 10,104–114.

Parducci, A. (1956). Category judgment: A range-frequency model. PsychologicalReview, 72, 407–418.

Peterson, W. W., Birdsall, T. G., & Fox, W. C. (1954). The theory of signal detectability.IRE trans. Professional group on information theory, 4, 171–212.

Pfanzagl, J. (1968). Theory of measurement. New York, NY: Wiley.Raaijmakers, J. G. W., & Shiffrin, R. M. (1981). Search of associative memory.

Psychological Review, 88, 93–134.Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85, 59–108.Regenwetter, M., & Marley, A. A. J. (2001). Random relations, random utilities, and

random functions. Journal of Mathematical Psychology, 45, 864–912.Restle, F., & Greeno, J. G. (1970). Introduction to mathematical psychology. Reading,

MA: Addison-Wesley Publishing Co.Roberts, F. S. (1979). Measurement theory with applications to decision making utility

and the social sciences. Reading, MA: Addison Wesley.Rumelhart, D., & McClelland, J. (1986). Parallel distributed processing: Explorations in

the microstructure of cognition: Vol. 1. Boston, MA: MIT Press.Sarris, V. (2006). Relational psychophysics in humans and animals: A comparative-

developmental approach. H & New York: Taylor & Francis Group.Schweickert, R., Giorgini, M., & Dzhafarov, E. (2000). Selective influence and re-

sponse time cumulative distribution functions in serial-parallel task networks.Journal of Mathematical Psychology, 44, 504–535.

Shepard, R. N. (1964). Attention and the metric structure of the stimulus space.Journal of Mathematical Psychology, 1, 54–87.

Smith, P. L., & Van Zandt, T. V. (2000). Time-dependent Poisson counter modelsof response latency in simple judgment. British Journal of Mathematical andStatistical Psychology, 53, 293–315.

Stevens, S. S. (1951). Mathematics, measurement and psychophysics. In S. S. Stevens(Ed.), Handbook of experimental psychology. New York, NY: Wiley.

Stout, J. C., Rock, S. L., Campbell, M. C., Busemeyer, J. R., & Finn, P. R. (2005).Psychological processes underlying risky decisions in drug abusers. Psychologyof Addictive Behaviors, 19(2), 148–157.

Suppes, P., Krantz, D. M., Luce, R. D., & Tversky, A. (1989). Foundations ofmeasurement: Geometrical threshold and probabilistic representations: Vol. 2. NewYork, NY: Academic Press.

Swets, J. A. (1996). Signal detection theory and ROC analysis in psychology anddiagnostics: Collected papers. Mahwah, NJ: Lawrence Erlbaum Associates.

Thomas, R. D. (1999). Assessing sensitivity in a multidimensional space: Someproblems and a definition of a general d’. Psychonomic Bulletin & Review, 6,224–238.

Thomas, R. D. (2003). Further considerations of a general d’ in multidimensionalspace. Journal of Mathematical Psychology, 47, 220–224.

Townsend, J.T. (1969). Moc parallel and serial models and experimental detectionof these. In Purdue Centennial Year Symposium on Information Processing (pp.617–625). Purdue University.

Townsend, J. T. (1971). A note on the identifiability of parallel and serial processes.Perception & Psychophysics, 10, 161–163.

Townsend, J. T. (1972). Some results concerning the identifiability of parallel andserial processes. British Journal of Mathematical and Statistical Psychology, 25,168–199.

Townsend, J. T. (1976a). A stochastic theory of matching processes. Journal ofMathematical Psychology, 14, 1–52.

Townsend, J. T. (1976b). Serial and within-stage independent parallel modelequivalence on the minimum completion time. Journal of MathematicalPsychology, 14, 219–238.

Townsend, J. T. (1994). A visual approach to nonlinear dynamics: Just cartoonsor a serious-pedagogieal device?. In Science Frontier Series, Review of A visualintroduction to dynamical systems theory for psychology by F.D. Abraham with R.H.Abraham & C.D. Shaw. Santa Cruz, CA: Aerial Press, 1991.

Townsend, J. T., Golden, R., & Wallsten, T. (2005). Quantitative training in psychologyis deteriorating: Traditional methodologists, mathematical psychologists, andpsychology face a challenge. Psychological Science Agenda, 19(9).

Townsend, J. T., & Kadlec, H. (1990). Mathematics and psychology. In R. E. Mickens(Ed.), Mathematics and science. Singapore: World Scientific Publishing Co.

Townsend, J. T., & Wenger, M. J. (2004). The serial parallel dilemma: A case study ina linkage of theory and method. Psychological Bulletin & Review, 11, 391–418.

Treat, T. A., McFall, R. M., Viken, R. J., & Kruschke, J. K. (2001). Using cognitive sciencemethods to assess the role of social information processing in sexually coercivebehavior. Psychological Assessment, 13, 549–565.

Uttal, W. R. (1998). Toward a new behaviorism: The case against perceptualreductionism. Mahwah, NJ: Lawrence Erlbaum Associates.

von Neumann, J., & Morgenstern, O. (1953). Theory of games and economic behavior.Princeton University Press.

Wagenmakers, E-J., & Waldorp, L. (Eds.) (2006). Model selection: Theoreticaldevelopments and applications. Journal of Mathematical Psychology, 50(2), Specialissue.

Weber, E. U., Shafir, S., & Blais, A. R. (2004). Predicting risk-sensitivity in humans andlower levels: Risk as variance or coefficient of variation. Psychological Review,111, 430–445.

Wenger, M. J., & Ingvalson, (2002). A decisional component of holistic encoding.Journal of Experimental Psychology: Learning, Memory and Cognition, 28, 872–892.

Wickens, T. D. (2002). Elementary signal detection theory. London: Oxford UniversityPress.

Wixted, J. T. (2007). Dual-process theory and signal-detection theory of recognitionmemory. Psychological Review, 114, 152–176.

Yonelinas, A. P. (1994). Receiver-operating characteristics in recognition memory:Evidence for a dual-process model. Journal of Experimental Psychology: Learning,Memory, & Cognition, 20, 1341–1354.

Journal of Mathematical Psychology Mathematical psychology

Documents

Transcript of Journal of Mathematical Psychology Mathematical psychology